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Mirrors > Home > NFE Home > Th. List > eqeqan12d | GIF version |
Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
eqeqan12d.1 | ⊢ (φ → A = B) |
eqeqan12d.2 | ⊢ (ψ → C = D) |
Ref | Expression |
---|---|
eqeqan12d | ⊢ ((φ ∧ ψ) → (A = C ↔ B = D)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeqan12d.1 | . 2 ⊢ (φ → A = B) | |
2 | eqeqan12d.2 | . 2 ⊢ (ψ → C = D) | |
3 | eqeq12 2365 | . 2 ⊢ ((A = B ∧ C = D) → (A = C ↔ B = D)) | |
4 | 1, 2, 3 | syl2an 463 | 1 ⊢ ((φ ∧ ψ) → (A = C ↔ B = D)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 = wceq 1642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-11 1746 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ex 1542 df-cleq 2346 |
This theorem is referenced by: eqeqan12rd 2369 adj11 3889 pw1equn 4331 pw1eqadj 4332 eqfnfv2 5393 pw1fnf1o 5855 dflec2 6210 tc11 6228 |
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